#216100

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#2856	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(2,5)(3,7)(9,10,12,15,13,16,11,14)(17,18)(19,26)(20,28,30,31)(21,25)(22,23)(24,27,32,29);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,28,32,31);(2,5)(6,8)(10,15)(11,12)(14,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,32)(26,28);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,16,14)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cei,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 123296);